-0.000 000 000 000 000 004 573 690 051 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 004 573 690 051 7(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
-0.000 000 000 000 000 004 573 690 051 7(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 000 000 000 000 004 573 690 051 7| = 0.000 000 000 000 000 004 573 690 051 7


2. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

3. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 004 573 690 051 7.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 000 000 004 573 690 051 7 × 2 = 0 + 0.000 000 000 000 000 009 147 380 103 4;
  • 2) 0.000 000 000 000 000 009 147 380 103 4 × 2 = 0 + 0.000 000 000 000 000 018 294 760 206 8;
  • 3) 0.000 000 000 000 000 018 294 760 206 8 × 2 = 0 + 0.000 000 000 000 000 036 589 520 413 6;
  • 4) 0.000 000 000 000 000 036 589 520 413 6 × 2 = 0 + 0.000 000 000 000 000 073 179 040 827 2;
  • 5) 0.000 000 000 000 000 073 179 040 827 2 × 2 = 0 + 0.000 000 000 000 000 146 358 081 654 4;
  • 6) 0.000 000 000 000 000 146 358 081 654 4 × 2 = 0 + 0.000 000 000 000 000 292 716 163 308 8;
  • 7) 0.000 000 000 000 000 292 716 163 308 8 × 2 = 0 + 0.000 000 000 000 000 585 432 326 617 6;
  • 8) 0.000 000 000 000 000 585 432 326 617 6 × 2 = 0 + 0.000 000 000 000 001 170 864 653 235 2;
  • 9) 0.000 000 000 000 001 170 864 653 235 2 × 2 = 0 + 0.000 000 000 000 002 341 729 306 470 4;
  • 10) 0.000 000 000 000 002 341 729 306 470 4 × 2 = 0 + 0.000 000 000 000 004 683 458 612 940 8;
  • 11) 0.000 000 000 000 004 683 458 612 940 8 × 2 = 0 + 0.000 000 000 000 009 366 917 225 881 6;
  • 12) 0.000 000 000 000 009 366 917 225 881 6 × 2 = 0 + 0.000 000 000 000 018 733 834 451 763 2;
  • 13) 0.000 000 000 000 018 733 834 451 763 2 × 2 = 0 + 0.000 000 000 000 037 467 668 903 526 4;
  • 14) 0.000 000 000 000 037 467 668 903 526 4 × 2 = 0 + 0.000 000 000 000 074 935 337 807 052 8;
  • 15) 0.000 000 000 000 074 935 337 807 052 8 × 2 = 0 + 0.000 000 000 000 149 870 675 614 105 6;
  • 16) 0.000 000 000 000 149 870 675 614 105 6 × 2 = 0 + 0.000 000 000 000 299 741 351 228 211 2;
  • 17) 0.000 000 000 000 299 741 351 228 211 2 × 2 = 0 + 0.000 000 000 000 599 482 702 456 422 4;
  • 18) 0.000 000 000 000 599 482 702 456 422 4 × 2 = 0 + 0.000 000 000 001 198 965 404 912 844 8;
  • 19) 0.000 000 000 001 198 965 404 912 844 8 × 2 = 0 + 0.000 000 000 002 397 930 809 825 689 6;
  • 20) 0.000 000 000 002 397 930 809 825 689 6 × 2 = 0 + 0.000 000 000 004 795 861 619 651 379 2;
  • 21) 0.000 000 000 004 795 861 619 651 379 2 × 2 = 0 + 0.000 000 000 009 591 723 239 302 758 4;
  • 22) 0.000 000 000 009 591 723 239 302 758 4 × 2 = 0 + 0.000 000 000 019 183 446 478 605 516 8;
  • 23) 0.000 000 000 019 183 446 478 605 516 8 × 2 = 0 + 0.000 000 000 038 366 892 957 211 033 6;
  • 24) 0.000 000 000 038 366 892 957 211 033 6 × 2 = 0 + 0.000 000 000 076 733 785 914 422 067 2;
  • 25) 0.000 000 000 076 733 785 914 422 067 2 × 2 = 0 + 0.000 000 000 153 467 571 828 844 134 4;
  • 26) 0.000 000 000 153 467 571 828 844 134 4 × 2 = 0 + 0.000 000 000 306 935 143 657 688 268 8;
  • 27) 0.000 000 000 306 935 143 657 688 268 8 × 2 = 0 + 0.000 000 000 613 870 287 315 376 537 6;
  • 28) 0.000 000 000 613 870 287 315 376 537 6 × 2 = 0 + 0.000 000 001 227 740 574 630 753 075 2;
  • 29) 0.000 000 001 227 740 574 630 753 075 2 × 2 = 0 + 0.000 000 002 455 481 149 261 506 150 4;
  • 30) 0.000 000 002 455 481 149 261 506 150 4 × 2 = 0 + 0.000 000 004 910 962 298 523 012 300 8;
  • 31) 0.000 000 004 910 962 298 523 012 300 8 × 2 = 0 + 0.000 000 009 821 924 597 046 024 601 6;
  • 32) 0.000 000 009 821 924 597 046 024 601 6 × 2 = 0 + 0.000 000 019 643 849 194 092 049 203 2;
  • 33) 0.000 000 019 643 849 194 092 049 203 2 × 2 = 0 + 0.000 000 039 287 698 388 184 098 406 4;
  • 34) 0.000 000 039 287 698 388 184 098 406 4 × 2 = 0 + 0.000 000 078 575 396 776 368 196 812 8;
  • 35) 0.000 000 078 575 396 776 368 196 812 8 × 2 = 0 + 0.000 000 157 150 793 552 736 393 625 6;
  • 36) 0.000 000 157 150 793 552 736 393 625 6 × 2 = 0 + 0.000 000 314 301 587 105 472 787 251 2;
  • 37) 0.000 000 314 301 587 105 472 787 251 2 × 2 = 0 + 0.000 000 628 603 174 210 945 574 502 4;
  • 38) 0.000 000 628 603 174 210 945 574 502 4 × 2 = 0 + 0.000 001 257 206 348 421 891 149 004 8;
  • 39) 0.000 001 257 206 348 421 891 149 004 8 × 2 = 0 + 0.000 002 514 412 696 843 782 298 009 6;
  • 40) 0.000 002 514 412 696 843 782 298 009 6 × 2 = 0 + 0.000 005 028 825 393 687 564 596 019 2;
  • 41) 0.000 005 028 825 393 687 564 596 019 2 × 2 = 0 + 0.000 010 057 650 787 375 129 192 038 4;
  • 42) 0.000 010 057 650 787 375 129 192 038 4 × 2 = 0 + 0.000 020 115 301 574 750 258 384 076 8;
  • 43) 0.000 020 115 301 574 750 258 384 076 8 × 2 = 0 + 0.000 040 230 603 149 500 516 768 153 6;
  • 44) 0.000 040 230 603 149 500 516 768 153 6 × 2 = 0 + 0.000 080 461 206 299 001 033 536 307 2;
  • 45) 0.000 080 461 206 299 001 033 536 307 2 × 2 = 0 + 0.000 160 922 412 598 002 067 072 614 4;
  • 46) 0.000 160 922 412 598 002 067 072 614 4 × 2 = 0 + 0.000 321 844 825 196 004 134 145 228 8;
  • 47) 0.000 321 844 825 196 004 134 145 228 8 × 2 = 0 + 0.000 643 689 650 392 008 268 290 457 6;
  • 48) 0.000 643 689 650 392 008 268 290 457 6 × 2 = 0 + 0.001 287 379 300 784 016 536 580 915 2;
  • 49) 0.001 287 379 300 784 016 536 580 915 2 × 2 = 0 + 0.002 574 758 601 568 033 073 161 830 4;
  • 50) 0.002 574 758 601 568 033 073 161 830 4 × 2 = 0 + 0.005 149 517 203 136 066 146 323 660 8;
  • 51) 0.005 149 517 203 136 066 146 323 660 8 × 2 = 0 + 0.010 299 034 406 272 132 292 647 321 6;
  • 52) 0.010 299 034 406 272 132 292 647 321 6 × 2 = 0 + 0.020 598 068 812 544 264 585 294 643 2;
  • 53) 0.020 598 068 812 544 264 585 294 643 2 × 2 = 0 + 0.041 196 137 625 088 529 170 589 286 4;
  • 54) 0.041 196 137 625 088 529 170 589 286 4 × 2 = 0 + 0.082 392 275 250 177 058 341 178 572 8;
  • 55) 0.082 392 275 250 177 058 341 178 572 8 × 2 = 0 + 0.164 784 550 500 354 116 682 357 145 6;
  • 56) 0.164 784 550 500 354 116 682 357 145 6 × 2 = 0 + 0.329 569 101 000 708 233 364 714 291 2;
  • 57) 0.329 569 101 000 708 233 364 714 291 2 × 2 = 0 + 0.659 138 202 001 416 466 729 428 582 4;
  • 58) 0.659 138 202 001 416 466 729 428 582 4 × 2 = 1 + 0.318 276 404 002 832 933 458 857 164 8;
  • 59) 0.318 276 404 002 832 933 458 857 164 8 × 2 = 0 + 0.636 552 808 005 665 866 917 714 329 6;
  • 60) 0.636 552 808 005 665 866 917 714 329 6 × 2 = 1 + 0.273 105 616 011 331 733 835 428 659 2;
  • 61) 0.273 105 616 011 331 733 835 428 659 2 × 2 = 0 + 0.546 211 232 022 663 467 670 857 318 4;
  • 62) 0.546 211 232 022 663 467 670 857 318 4 × 2 = 1 + 0.092 422 464 045 326 935 341 714 636 8;
  • 63) 0.092 422 464 045 326 935 341 714 636 8 × 2 = 0 + 0.184 844 928 090 653 870 683 429 273 6;
  • 64) 0.184 844 928 090 653 870 683 429 273 6 × 2 = 0 + 0.369 689 856 181 307 741 366 858 547 2;
  • 65) 0.369 689 856 181 307 741 366 858 547 2 × 2 = 0 + 0.739 379 712 362 615 482 733 717 094 4;
  • 66) 0.739 379 712 362 615 482 733 717 094 4 × 2 = 1 + 0.478 759 424 725 230 965 467 434 188 8;
  • 67) 0.478 759 424 725 230 965 467 434 188 8 × 2 = 0 + 0.957 518 849 450 461 930 934 868 377 6;
  • 68) 0.957 518 849 450 461 930 934 868 377 6 × 2 = 1 + 0.915 037 698 900 923 861 869 736 755 2;
  • 69) 0.915 037 698 900 923 861 869 736 755 2 × 2 = 1 + 0.830 075 397 801 847 723 739 473 510 4;
  • 70) 0.830 075 397 801 847 723 739 473 510 4 × 2 = 1 + 0.660 150 795 603 695 447 478 947 020 8;
  • 71) 0.660 150 795 603 695 447 478 947 020 8 × 2 = 1 + 0.320 301 591 207 390 894 957 894 041 6;
  • 72) 0.320 301 591 207 390 894 957 894 041 6 × 2 = 0 + 0.640 603 182 414 781 789 915 788 083 2;
  • 73) 0.640 603 182 414 781 789 915 788 083 2 × 2 = 1 + 0.281 206 364 829 563 579 831 576 166 4;
  • 74) 0.281 206 364 829 563 579 831 576 166 4 × 2 = 0 + 0.562 412 729 659 127 159 663 152 332 8;
  • 75) 0.562 412 729 659 127 159 663 152 332 8 × 2 = 1 + 0.124 825 459 318 254 319 326 304 665 6;
  • 76) 0.124 825 459 318 254 319 326 304 665 6 × 2 = 0 + 0.249 650 918 636 508 638 652 609 331 2;
  • 77) 0.249 650 918 636 508 638 652 609 331 2 × 2 = 0 + 0.499 301 837 273 017 277 305 218 662 4;
  • 78) 0.499 301 837 273 017 277 305 218 662 4 × 2 = 0 + 0.998 603 674 546 034 554 610 437 324 8;
  • 79) 0.998 603 674 546 034 554 610 437 324 8 × 2 = 1 + 0.997 207 349 092 069 109 220 874 649 6;
  • 80) 0.997 207 349 092 069 109 220 874 649 6 × 2 = 1 + 0.994 414 698 184 138 218 441 749 299 2;
  • 81) 0.994 414 698 184 138 218 441 749 299 2 × 2 = 1 + 0.988 829 396 368 276 436 883 498 598 4;
  • 82) 0.988 829 396 368 276 436 883 498 598 4 × 2 = 1 + 0.977 658 792 736 552 873 766 997 196 8;
  • 83) 0.977 658 792 736 552 873 766 997 196 8 × 2 = 1 + 0.955 317 585 473 105 747 533 994 393 6;
  • 84) 0.955 317 585 473 105 747 533 994 393 6 × 2 = 1 + 0.910 635 170 946 211 495 067 988 787 2;
  • 85) 0.910 635 170 946 211 495 067 988 787 2 × 2 = 1 + 0.821 270 341 892 422 990 135 977 574 4;
  • 86) 0.821 270 341 892 422 990 135 977 574 4 × 2 = 1 + 0.642 540 683 784 845 980 271 955 148 8;
  • 87) 0.642 540 683 784 845 980 271 955 148 8 × 2 = 1 + 0.285 081 367 569 691 960 543 910 297 6;
  • 88) 0.285 081 367 569 691 960 543 910 297 6 × 2 = 0 + 0.570 162 735 139 383 921 087 820 595 2;
  • 89) 0.570 162 735 139 383 921 087 820 595 2 × 2 = 1 + 0.140 325 470 278 767 842 175 641 190 4;
  • 90) 0.140 325 470 278 767 842 175 641 190 4 × 2 = 0 + 0.280 650 940 557 535 684 351 282 380 8;
  • 91) 0.280 650 940 557 535 684 351 282 380 8 × 2 = 0 + 0.561 301 881 115 071 368 702 564 761 6;
  • 92) 0.561 301 881 115 071 368 702 564 761 6 × 2 = 1 + 0.122 603 762 230 142 737 405 129 523 2;
  • 93) 0.122 603 762 230 142 737 405 129 523 2 × 2 = 0 + 0.245 207 524 460 285 474 810 259 046 4;
  • 94) 0.245 207 524 460 285 474 810 259 046 4 × 2 = 0 + 0.490 415 048 920 570 949 620 518 092 8;
  • 95) 0.490 415 048 920 570 949 620 518 092 8 × 2 = 0 + 0.980 830 097 841 141 899 241 036 185 6;
  • 96) 0.980 830 097 841 141 899 241 036 185 6 × 2 = 1 + 0.961 660 195 682 283 798 482 072 371 2;
  • 97) 0.961 660 195 682 283 798 482 072 371 2 × 2 = 1 + 0.923 320 391 364 567 596 964 144 742 4;
  • 98) 0.923 320 391 364 567 596 964 144 742 4 × 2 = 1 + 0.846 640 782 729 135 193 928 289 484 8;
  • 99) 0.846 640 782 729 135 193 928 289 484 8 × 2 = 1 + 0.693 281 565 458 270 387 856 578 969 6;
  • 100) 0.693 281 565 458 270 387 856 578 969 6 × 2 = 1 + 0.386 563 130 916 540 775 713 157 939 2;
  • 101) 0.386 563 130 916 540 775 713 157 939 2 × 2 = 0 + 0.773 126 261 833 081 551 426 315 878 4;
  • 102) 0.773 126 261 833 081 551 426 315 878 4 × 2 = 1 + 0.546 252 523 666 163 102 852 631 756 8;
  • 103) 0.546 252 523 666 163 102 852 631 756 8 × 2 = 1 + 0.092 505 047 332 326 205 705 263 513 6;
  • 104) 0.092 505 047 332 326 205 705 263 513 6 × 2 = 0 + 0.185 010 094 664 652 411 410 527 027 2;
  • 105) 0.185 010 094 664 652 411 410 527 027 2 × 2 = 0 + 0.370 020 189 329 304 822 821 054 054 4;
  • 106) 0.370 020 189 329 304 822 821 054 054 4 × 2 = 0 + 0.740 040 378 658 609 645 642 108 108 8;
  • 107) 0.740 040 378 658 609 645 642 108 108 8 × 2 = 1 + 0.480 080 757 317 219 291 284 216 217 6;
  • 108) 0.480 080 757 317 219 291 284 216 217 6 × 2 = 0 + 0.960 161 514 634 438 582 568 432 435 2;
  • 109) 0.960 161 514 634 438 582 568 432 435 2 × 2 = 1 + 0.920 323 029 268 877 165 136 864 870 4;
  • 110) 0.920 323 029 268 877 165 136 864 870 4 × 2 = 1 + 0.840 646 058 537 754 330 273 729 740 8;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


5. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 004 573 690 051 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1110 1001 0001 1111 0110 0010 11(2)

6. Positive number before normalization:

0.000 000 000 000 000 004 573 690 051 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1110 1001 0001 1111 0110 0010 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 58 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 004 573 690 051 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1110 1001 0001 1111 0110 0010 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 1010 0011 1111 1110 1001 0001 1111 0110 0010 11(2) × 20 =

1.0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011(2) × 2-58


8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 1 (a negative number)

Exponent (unadjusted): -58

Mantissa (not normalized):
1.0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-58 + 2(11-1) - 1 =

(-58 + 1 023)(10) =

965(10)


10. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 965 ÷ 2 = 482 + 1;
  • 482 ÷ 2 = 241 + 0;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

965(10) =

011 1100 0101(2)


12. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011 =

0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1100 0101

Mantissa (52 bits) =
0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011


Decimal number -0.000 000 000 000 000 004 573 690 051 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 0101 - 0101 0001 0111 1010 1000 1111 1111 1010 0100 0111 1101 1000 1011

» Convert decimal -0.000 000 000 000 000 004 573 690 045 8 to 64 bit double precision IEEE 754 binary floating point representation standard

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100